Research Seminar: Geometric aspects of the Ginzburg-Landau equations

Speaker: Robert Jerrard
Title: The Ginzburg-Landau equations with vorticity concentrating near a non-degenerate geodesic

Abstract: It is well-known that under suitable hypotheses, for a sequence of solutions of the (simplified) Ginzburg-Landau equations $-\Delta u_\epsilon +\epsilon^{-2}(|u_\epsilon|^2-1)u_\epsilon = 0$, the energy and vorticity concentrate as $\epsilon\to 0$ around a codimension $2$ stationary varifold --- a (measure theoretic) minimal surface. Much less is known about the question of whether, given a codimension $2$ minimal surface, there exists a sequence of solutions  for  which the given minimal surface is the limiting concentration set.  The corresponding question is very well-understood for minimal hypersurfaces and the scalar Allen-Cahn equation, and for the Ginzburg-Landau equations when the minimal surface is locally area-minimizing, but otherwise quite open.

We consider this question on a $3$-dimensional closed Riemannian manifold $(M,g)$, and we prove that any embedded nondegenerate closed geodesic can be realized as the asymptotic energy/vorticity concentration set of a sequence of solutions of the Ginzburg-Landau equations.